关于高阶导数拉格朗日理论中的广义力
On generalized forces in higher derivative Lagrangian theory
摘要：
在这篇文章中，我们介绍了这种形式的$α_1 A_μ(x)\dot{x}^μ$，$α_2 G_μ(x)\ddot{x}^μ$，$α_3 B_μ(x)\dddot{x}^μ$，$α_4 K_μ(x)\ddddot{x}^μ$，$\cdots$的高阶导数拉格朗日量，它将电磁相互作用推广到高阶导数。我们表明，奇数阶拉格朗日描述的相互作用类似于电磁，而偶数阶拉格朗日类似于引力相互作用。从这个类比，我们制定了广义感应原理的概念，假设更高的场$U_{(n),μ}(x),\ n\geq1$和更高的电流$j^{(n)μ}=ρ(x)d^nx^μ/ds^n$之间的耦合，其中$ρ(x)$是质量（$n$偶数）或电荷（$n$奇数）的空间密度。简而言之，这篇文章是一个邀请，反思力和惯性概念的概括。我们将在本文的最后一节更深入地讨论这些范式的含义。
英文摘要：
In this article, we introduce higher derivative Lagrangians of this form $α_1 A_μ(x)\dot{x}^μ$, $α_2 G_μ(x)\ddot{x}^μ$, $α_3 B_μ(x)\dddot{x}^μ$, $α_4 K_μ(x)\ddddot{x}^μ$, $\cdots$, that generalize the electromagnetic interaction to higher order derivatives. We show that odd order Lagrangians describe interactions analog to electromagnetism while even order Lagrangians are similar to gravitational interaction. From this analogy, we formulate the concept of the generalized induction principle assuming the coupling between the higher fields $U_{(n),μ}(x),\ n\geq1$ and the higher currents $j^{(n)μ}=ρ(x)d^nx^μ/ds^n$, where $ρ(x)$ is the spatial density of mass ($n$ even) or of electric charge ($n$ odd). In short, this article is an invitation to reflect on a generalization of the concept of force and of inertia. We discuss the implications of these paradigms more in depth in the last section of the paper.
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